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# A certain square is to be drawn on a coordinate plane. One

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Senior Manager
Joined: 21 Jun 2006
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A certain square is to be drawn on a coordinate plane. One [#permalink]  16 Jun 2007, 20:53
A certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn?
Director
Joined: 14 Jan 2007
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12 ways the square can be drawn.
there will be three squares in each quadrant with the vertices other than origin - (10,0), (0,10), (10,10) -- 1
(6,8), (-8,6) , (-2,14) --2
(8,6), (-6,8), (2,14) --3
VP
Joined: 08 Jun 2005
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I figured 4 ways

(0,0) - (0,10) - (10,0) - (10,10) = 100

(0,0) - (10,0) - (0,-10) - (-10,-10) = 100

(0,0) - (0,-10) - (10,0) - (10,10) = 100

(0,0) - (0,-10) - (-10,0) - (-10,-10) = 100

VP
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Re: coordinate plane [#permalink]  17 Jun 2007, 13:21
ArvGMAT wrote:
A certain square is to be drawn on a coordinate plane. One of the vertices must be on the origin, and the square is to have an area of 100. If all coordinates of the vertices must be integers, how many different ways can this square be drawn?

I would guess 8 ways total, but I am not sure

Set the the lower right vertex of the square to be (a,b) we know that for quadrant 1 and 2,
a^2 + b^2 = 100

Possible integers for (a,b): (10,0), (8,6), (6,8), (0,10)

Therefore, 4 quadrants would give 8.
I don't know how to check the other points...
Director
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vshaunak@gmail.com wrote:
12 ways the square can be drawn.
there will be three squares in each quadrant with the vertices other than origin - (10,0), (0,10), (10,10) -- 1
(6,8), (-8,6) , (-2,14) --2
(8,6), (-6,8), (2,14) --3

12:

I: (10, 0), (8, 6), and (6, 8)
I: (0, 10), (-6, 8), and (-8, 6)
I: (-10, 0), (-8, -6), and (-6, -8)
I: (-10, 0), (8, -6), and (6, -8)
Senior Manager
Joined: 21 Jun 2006
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OA is 8.
here are the combos -
10,0
0,10
-10,0
0,-10
6,8
-8,-6
-6,8
-8,6
Manager
Joined: 28 Aug 2006
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ArvGMAT,

Would you give an example as how a square is formed using the below numbers. Thank you

6,8
-8,-6
-6,8
-8,6
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